Page 41 - Determinants and Their Applications in Mathematical Physics

26 3. Intermediate Determinant Theory

Recalling the definitions of rejecter minors M, retainer minors N, and
cofactors A, each with row and column parameters, it is found that

,
y i 1 ··· y i r = N i 1 ...i r ;j 1 ...j r e j 1 ··· e j r
∗ ∗
e 1 ··· e n ,
z 1 ··· z n = M i 1 ...i r ;j 1 ...j r
where, in this case, the symbol ∗ denotes that those vectors with suffixes
j 1 ,j 2 ,...,j r are omitted. Hence,
x 1 ··· x n

∗
= (−1) N i 1 i 2 ...i r ;j 1 j 2 ...j r M i 1 i 2 ...,i r ;j 1 j 2 ...j r e j 1 ··· e j r e 1 ··· e n .
p
i 1 ...i r
By applying in reverse order the sequence of interchanges used to obtain
(3.3.2), it is found that
∗
e 1 ··· e n =(−1) (e 1 ··· e n ),
q
e j 1 ··· e j r
where
n
1
q = j s − r(r +1).
2
s=1
Hence,

x 1 ··· x n = (−1) p+q N i 1 i 2 ...i r ;j 1 j 2 ...j r M i 1 i 2 ...i r ;j 1 j 2 ...j r e 1 ··· e n
i 1 ...i r

= N i 1 i 2 ...i r ;j 1 j 2 ...j r A i 1 i 2 ...i r ;j 1 j 2 ...j r e 1 ··· e n .
i 1 ...i r
Comparing this formula with (1.2.5) in the section on the definition of a
determinant, it is seen that

A n = |a ij | n = N i 1 i 2 ...i r ;j 1 j 2 ...j r A i 1 i 2 ...i r ;j 1 j 2 ...j r , (3.3.4)
i 1 ...i r
which is the general form of the Laplace expansion of A n in which the sum
extends over the row parameters. By a similar argument, it can be shown
that A n is also equal to the same expression in which the sum extends over
the column parameters.
When r = 1, the Laplace expansion degenerates into a simple expansion
by elements from column j or row i and their first cofactors:

A n = N ij A ij ,
i or j

= a ij A ij .
i or j